Optimal. Leaf size=136 \[ -\frac {3 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}+\frac {4 x^{1+m} \sqrt {c-a^2 c x^2} \, _2F_1(1,1+m;2+m;a x)}{(1+m) \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.14, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6288, 6285, 90,
66} \begin {gather*} \frac {4 x^{m+1} \sqrt {c-a^2 c x^2} \, _2F_1(1,m+1;m+2;a x)}{(m+1) \sqrt {1-a^2 x^2}}-\frac {3 x^{m+1} \sqrt {c-a^2 c x^2}}{(m+1) \sqrt {1-a^2 x^2}}-\frac {a x^{m+2} \sqrt {c-a^2 c x^2}}{(m+2) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 90
Rule 6285
Rule 6288
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} x^m \sqrt {c-a^2 c x^2} \, dx &=\frac {\sqrt {c-a^2 c x^2} \int e^{3 \tanh ^{-1}(a x)} x^m \sqrt {1-a^2 x^2} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \frac {x^m (1+a x)^2}{1-a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {c-a^2 c x^2} \int \left (-3 x^m-a x^{1+m}+\frac {4 x^m}{1-a x}\right ) \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}+\frac {\left (4 \sqrt {c-a^2 c x^2}\right ) \int \frac {x^m}{1-a x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {3 x^{1+m} \sqrt {c-a^2 c x^2}}{(1+m) \sqrt {1-a^2 x^2}}-\frac {a x^{2+m} \sqrt {c-a^2 c x^2}}{(2+m) \sqrt {1-a^2 x^2}}+\frac {4 x^{1+m} \sqrt {c-a^2 c x^2} \, _2F_1(1,1+m;2+m;a x)}{(1+m) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 74, normalized size = 0.54 \begin {gather*} -\frac {x^{1+m} \sqrt {c-a^2 c x^2} (6+a x+m (3+a x)-4 (2+m) \, _2F_1(1,1+m;2+m;a x))}{(1+m) (2+m) \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (a x +1\right )^{3} x^{m} \sqrt {-a^{2} c \,x^{2}+c}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{m} \sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m\,\sqrt {c-a^2\,c\,x^2}\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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